GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Using Genetic Algorithms to solve the
Minimum Labeling Spanning Tree Problem
Final Presentation
Oliver Rourke, [email protected]
Advisor: Dr Bruce L. Golden, [email protected]. H. Smith School of Business
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 1 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Introduction to Minimum Labelling Spanning TreeProblem (MLST)
Combinatorial optimization problem first proposed in 1996[Chang:1996]
Connected Graph - set of vertices and edges.
Each edge has a label
Find the smallest set of labels which gives a connectedsub-graph
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 2 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
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Parallel GAs
Software/Validationetc...
An example of a labelled spanning tree, andsubgraphs
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GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
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More about MLST
NP-complete - ’perfect’ algorithm impossible (?)
Many heuristics have been used including:
Variable Neighborhood Search (VNS) - BestSimulated AnnealingPilot MethodReactive Tabu Search
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 4 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Introduction to Genetic Algorithms (GAs)
Evolutionary-inspired heuristic for optimization problems
Population = set of (valid) solutions
Select, Breed, Replace
Advantages:
Flexible and adaptableRobust performance at global searchSimple to parallelize
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 5 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Introduction to Genetic Algorithms (GAs)
Evolutionary-inspired heuristic for optimization problems
Population = set of (valid) solutions
Select, Breed, Replace
Advantages:
Flexible and adaptableRobust performance at global searchSimple to parallelize
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 5 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Introduction to Genetic Algorithms (GAs)
Evolutionary-inspired heuristic for optimization problems
Population = set of (valid) solutions
Select, Breed, Replace
Advantages:
Flexible and adaptableRobust performance at global searchSimple to parallelize
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 5 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
One Parameter GA for MLST - Serial
From Xiong, 2005
Designed to be simple - no fine tuning
One parameter - p, population size
Solution: List of labels (gives connected sub-graph)
Gene: Label in the list
Modified Genetic Algorithm (MGA), Xiong, 2006 - moreintelligent crossover operator
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 6 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
GA: Overview
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 7 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
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GA Improvements
1: Coin toss: Make crossover/mutation stochastic
2: Keep equally fit offspring over parents
3: Favor mutation: Encourage retention of new material
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 8 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Databases
Two distinct databases:
36 sets of instances (10 instances per set) from Cerulli etal. [2005], smaller graphs. Used for validation only.
5 sets of 100 randomly generated larger graphs (usingtechnique from Xiong, 2005). Each set has 100 nodes, 0.2edge density and either 25, 50, 100, 250 or 500 labels.
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 9 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Serial Testing
Conducted on Genome cluster at UMD
All experiments using one processor, instances runsequentially
Tested on own generated databases (same as above) withN = 100, p = 0.2, L = 25, 50, 100, 250, 500
Run 10 times with different random number seeds
Stop: Max running time (L*20ms per instance) vs. Maxgeneration count
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 10 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Serial GA changes - results
GA + variants:Algorithm % Above % Above Optimum Time
BKS (if known) (s)Original GA 6.97 3.79 129.3
Crossover Coin toss 4.7 2.3 139.8Mutation Coin Toss 5.1 3.0 130.5
Keep Equal 8.3 5.1 123.1Favor Mutation 5.1 2.4 130.11
Everything 3.9 2.4 139.7
Xiong’s Modified GA (MGA) + variantsAlgorithm % Above % Above Optimum Time
BKS (if known) (s)MGA 4.1 2.2 474.8
MGA with Everything 2.9 1.5 421.1
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 11 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Results vs Iteration count (L=50)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 12 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Results vs Time (L=50)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 13 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
All Results vs Time
(a) L=25 (b) L=50
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 14 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
All Results vs Time
(c) L=100 (d) L=250
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 15 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
All Results vs Time
(e) L=500
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 16 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Serial Algorithm − > Parallel Algorithm
Why?
SpeedupLarger ProblemsEffectively use all available computational resources
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 17 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Dividing up the Population
Figure: Three different types of GAs showing interaction betweenindividuals (black dots) in the population. a)Panmictic b) Distributedc) Cellular [Alba:2008]
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 18 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Distributed GA - results
Figure: % above BKS and computational time for a variety of islandsizes - no migration (run on 2.2GHz quad-core Intel Core i7)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 19 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Serial Algorithm − > Parallel Algorithm
Allocate different subpopulations to different processors
Communication between subpopulations - better results?
Master-slave versus Direct communication (Who?)
Message Passing versus Shared Memory (How?)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 20 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Serial Algorithm − > Parallel Algorithm
Allocate different subpopulations to different processors
Communication between subpopulations - better results?
Master-slave versus Direct communication (Who?)
Message Passing versus Shared Memory (How?)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 20 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Parallel Structures (Who?)
(a) Master-slave (b) Direct communication
Figure: Different approaches to parallel programming
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 21 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
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Software/Validationetc...
Communication Schemes(How?)
(a) Message Passing (b) Shared Memory
Figure: Different approaches to inter-processor communication
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 22 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Local communication
Arrange subpopulations on grid, define neighborhood ongrid [Scharrenbroich: ’CGA-inspired’ distributed GA]
Carry strongest individuals between ’neighboring’subpopulations at certain points in algorithm
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 23 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Mesh diagrams
(a) One dimensional (b) Two dimensional
Figure: Different mesh arrangement with one possible neighborhooddefinition in (b)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 24 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Local communication
Arrange subpopulations on grid, define neighborhood ongrid [Scharrenbroich: ’CGA-inspired’ distributed GA]
Carry strongest individuals between ’neighboring’subpopulations at certain points in algorithm
How?
Replace weakest individuals locally?Place in ’waiting room’ where they can be accessed, notdirectly replacing...
When?
Regular intervals?When population stagnates
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 25 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
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Software/Validationetc...
Direct Communication results
At best: negligible improvement
Why? Problem, population size, number of processors...
OR Straight up bad idea (in this case)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 26 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Direct Communication results
At best: negligible improvement
Why? Problem, population size, number of processors...
OR Straight up bad idea (in this case)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 26 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Global Communication
All subpopulations connected through common ’vault’
Strongest unique solutions found to date stored in vault
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 27 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
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Parallel GAs
Software/Validationetc...
Vault Diagram
Figure: 1D Mesh with vault included
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 28 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Vault implications
Communication from subpopulations to vault - best,unique individuals
Communication out of vault - occasionally breed with arandomly selected individual out of vault (modifyselection)
Evolution can lose local optima - vault will maintain globaloptima
Simulated Annealing type selectionRespawning (Individual?Subpopulation?)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 29 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Vault implications
Communication from subpopulations to vault - best,unique individuals
Communication out of vault - occasionally breed with arandomly selected individual out of vault (modifyselection)
Evolution can lose local optima - vault will maintain globaloptima
Simulated Annealing type selectionRespawning (Individual?Subpopulation?)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 29 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Vault implications
Communication from subpopulations to vault - best,unique individuals
Communication out of vault - occasionally breed with arandomly selected individual out of vault (modifyselection)
Evolution can lose local optima - vault will maintain globaloptima
Simulated Annealing type selectionRespawning (Individual?Subpopulation?)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 29 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Parallel Testing
Conducted on Genome cluster at UMD
All experiments involving 32 threads with 32 processorsand 32 subpopulations (or separate VNS trials)
Tested on own generated databases (same as above) withN = 100, p = 0.2, L = 25, 50, 100, 250, 500
Run 10 times with different random number seeds
Max running time = L*20ms per instance (for eachprocessor)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 30 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
All Results vs Time
(a) L=25 (b) L=50
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 31 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
All Results vs Time
(c) L=100 (d) L=250
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 32 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
All Results vs Time
(e) L=500
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 33 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Vault extensions
Vault evolution (separate exploration of search space withinvestigation of interesting areas)
Replace unique with different enough (Hamming orLevenshtein distance...)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 34 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Vault extensions
Vault evolution (separate exploration of search space withinvestigation of interesting areas)
Replace unique with different enough (Hamming orLevenshtein distance...)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 34 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
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Software/Hardware
C++ with pthreads
Run on Genome cluster at UMD - Quad-Core AMDOpteron R© Processor 8382 (2.6GHz)
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GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Serial Validation
Genetic Algorithm: Compared with original code fromXiong et al. [2005]. When random seed set correctly andrandom numbers sampled in the same order, returned thesame results.
VNS: Results compared with the results reported inConsoli (2009). Similar results achieved (no statisticallysignificant difference)
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 36 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Parallel Validation
Remove all inter-processor communication, record resultsfrom each processor individually. Check similar to serialcode.
Verify the sending and receiving for each type of messageon both ends
Investigate speed-up (on 32 processor machine, 32subpopulations for parallel code):
Instance Set L=100 L=500
Time for serial (s) 2.38 5.42Parallel-No Comm (s) 3.75 8.25
Parallel-Synchronized Iterations (s) 4.35 10.89Parallel-Synchronized+Vault (s) 4.51 12.03
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 37 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Goals
Create own competitive, efficient, serial GA code
Convert to an efficient parallel GA, first synchronous andlater asynchronous.
Fine tune parallel GA (investigate migration operators)
Run optimized code on large array of processors
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 38 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Goals
Create own competitive, efficient, serial GA code
Convert to an efficient parallel GA, first synchronous andlater asynchronous.
Fine tune parallel GA (investigate migration operators)
Run optimized code on large array of processors
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 38 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Goals
Create own competitive, efficient, serial GA code
Convert to an efficient parallel GA, first synchronous andlater asynchronous.
Fine tune parallel GA (investigate migration operators)
Run optimized code on large array of processors
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 38 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Goals
Create own competitive, efficient, serial GA code
Convert to an efficient parallel GA, first synchronous andlater asynchronous.
Fine tune parallel GA (investigate migration operators)
Run optimized code on large array of processors
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 38 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Deliverables
Efficient, competitive serial GA code for the MLST
Efficient, asynchronous and synchronous parallel GA codefor the MLST
Results from running code on appropriate machines
Report, presentation
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 39 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Thanks
A sincere thank you to Dr Golden for so much invaluableadvice, patient listening and watching over the whole project
Thanks also go to Drs Ide and Balan for help with presentingwork and other advice, to Dr Zimin for help with the Genomecluster and many pointers to do with parallel computing, andto all of you for listening and for your questions
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 40 / 42
GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
GeneticAlgorithms
My SerialGAs
Parallel GAs
Software/Validationetc...
Bibliography
Alba, E. and Dorronsoro, B., Cellular Genetic Algorithms, Springer, NY, 2008
Back, T., Hammel., U and Schwefel, H., Evolutionary Computation: Comments on the History andCurrent State, IEEE Transactions on evolutionary computation, Vo. 1, No 1, 1997
Canyurt, O. And Hajela, P., Cellular Genetic algorithm technique for the multicriterion designoptimization, Struct. Multidisc Optim 20, 2010
Chang, R. and Leu, S., The minimum labeling spanning trees, Information Processing Letters 63,1997
Cerulli, R. et al., Metaheuristics comparison for the minimum labelling spanning tree problem in TheNext Wave in Computing, Optimization, and Decision Technologies, vol. 29 of OperationsResearch/Computer Science Interfaces Series, NY, NY, USA, 2005
Consoli, S. et al., Greedy Randomized Adaptive Search and Variable Neighbourhood Search for theminimum labelling spanning tree problem, European Journal of Operational Research, vol. 196 pp.440-449, 2009
Drummond, L., Ochi, L. and Vianna, D., An Asynchronous parallel metaheuristic for the periodvehicle routing problem, Future Generation Computer Systems 17, 2001
Groer, C., Golden, B. and Wasil, E., A Parallel Algorithm for the Vehicle Routing Problem,INFORMS Journal on Computing, 2010
Fujimoto, N. and Tsutsui, S., A Highly-Parallel TSP Solver for a GPU Computing Platform, LNCS6046, 2011
Huy, N. et al., Adaptive Cellular Memttic Algorithms, Evolutionary Computation 17(2), 2009
Karova, M., Smarkov, V. and Penev, S, Genetic operators crossover and mutation in solving the TSPproblem, International conference on computer systems and technologies, 2005. Katayama,Hirabayashi, Naruhusa, Performance Analysis for Crossover Operators of Genetic Algorithm, Systemsand Computers in Japan, Vol 30., No 2., 1999
Papaioannou, G. and Wilson, J., The evolution of cell formation problem methodologies based onrecent studies (1997-2008): Review and directions for future research, European Journal ofOperational Research 206, 2010
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GeneticAlgorithm forthe MLST
Oliver Rourke
The MLST
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Bibliography (cont.)
Paszkowicz, W., Properties of a Genetic algorithm extended by a random self-learning operator andasymmetric mutations: A convergence study for a task of powder-pattern indexing, AnalyticsChimica Acta 566 (2006)
Sarma J., and De Jong, K., An analysis of the effect of the neighborhood size and shape on localselection algorithms. In H.M. Voigt, W. Ebeling, I. Rechenberg, and H.P. Schwefel, editors, Proc. ofthe International Confer- ence on Parallel Problem Solving from Nature IV (PPSN-IV), volume 1141of Lecture Notes in Computer Science (LNCS), pages 236244. Springer-Verlag, Heidelberg, 1996.
Simoncini D., et al., From Cells to Islands: An Unified Model of Cellular Parallel Genetic Algorithms,ACRI, 2006.
Seredynski, F. and Zomaya, A., Sequential and Parallel Automata-Based Scheduling Algorithms, IEETransactions on Parallel and Distributed Systems, Vol 13, No 10, 2002
Serpell, M. and Smith, E., Self-Adaptation of Mutation Operator and Probability for PermutationRepresentation in Genetic Algorithms, Evolutionary Computation 18(3), 2010
Vidal, P. and Alba, E., A Multi-GPU Implementation of a Cellular Genetic Algorithm, IEEE 2010
Xiong, Y., Golden, B. and Wasil, E., A One-Parameter Genetic Algorithm for the Minimum labelingSpanning Tree problem, IEEE Transactions on evolutionary computation, Vol. 9, No. 1, 2005
Xiong, Y., Golden, B., and Wasil, E., Improved Heuristic for the Minimum Label Spanning TreeProblem, IEEE Transactions on evolutionary computing, Vol 10., No. 6, 2006
Oliver Rourke (UMD) Genetic Algorithm for the MLST May 3, 2012 42 / 42